Abstract
MV-algebras with internal states, or SMV-algebras for short, are the equivalent algebraic semantics of the logic \(\mathrm{SFP}(\mathrm{\L },\mathrm{\L })\) which allows to represent and reason about the probability of infinite-valued events. In this paper we will make the first steps towards establishing completeness for \(\mathrm{SFP}(\mathrm{\L },\mathrm{\L })\) with respect to the class of standard SMV-algebras, a problem which has been left open since the first paper on SMV-algebras was published. More precisely we will prove that, if we restrict our attention to a particular, yet expressive, subclass of formulas, then theorems of \(\mathrm{SFP}(\mathrm{\L },\mathrm{\L })\) are the same as tautologies of a class of SMV-algebras that can be reasonably called “standard”.
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References
Chang, C.C.: Algebraic analysis of many-valued logics. Trans. Am. Math. Soc. 88, 467–490 (1958)
Cignoli, R., D’Ottaviano, I.M.L., Mundici, D.: Algebraic Foundationsof Many-valued Reasoning. Kluwer, Alphen aan den Rijn (2000)
Flaminio, T., Godo, L.: A logic for reasoning about the probability of fuzzy events. Fuzzy Sets Syst. 158(6), 625–638 (2007)
Flaminio, T., Hosni, H., Lapenta, S.: Convex MV-algebras: Many-valued logics meet decision theory. Stud. Logica 106(5), 913–945 (2018)
Flaminio, T., Kroupa, T.: States of MV-algebras. Chapter XVII of Handbook of Mathematical Fuzzy Logic - volume 3. In: Fermüller, C., Cintula, P., Noguera, C. (Eds.) Studies in Logic, Mathematical Logic and Foundations, vol. 58. College Publications, London (2015)
Flaminio, T., Montagna, F.: MV-algebras with internal states and probabilistic fuzzy logics. Int. J. Approximate Reasoning 50(1), 138–152 (2009)
Flaminio, T., Montagna, F.: Models for many-valued probabilistic reasoning. J. Logic Comput. 21(3), 447–464 (2011)
Hájek, P.: Metamathematics of Fuzzy Logics. Kluwer Academic Publishers, Dordrecht (1998)
Horn, A., Tarski, A.: Measures on Boolean algebras. Trans. Am. Math. Soc. 64, 467–497 (1948)
Kroupa, T.: Every state on semisimple MV-algebra is integral. Fuzzy Sets Syst. 157(20), 2771–2787 (2006)
Kroupa, T.: Representation and extension of states on MV-algebras. Arch. Math. Logic 45, 381–392 (2006)
Mundici, D.: Averaging the truth-value in Łukasiewicz logic. Stud. Logica 55, 113–127 (1995)
Mundici, D.: Tensor products and the Loomis-Sikorski theorem for MV-algebras. Adv. Appl. Math. 22, 227–248 (1999)
Panti, G.: Invariant measures on free MV-algebras. Commun. Algebra 36(8), 2849–2861 (2009)
Acknowledgments
The author acknowledges partial support by the Spanish Ramon y Cajal research program RYC-2016-19799; the Spanish FEDER/MINECO project TIN2015-71799-C2-1-P and the SYSMICS project (EU H2020-MSCA-RISE-2015, Project 689176).
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Flaminio, T. (2019). Towards a Standard Completeness for a Probabilistic Logic on Infinite-Valued Events. In: Kern-Isberner, G., Ognjanović, Z. (eds) Symbolic and Quantitative Approaches to Reasoning with Uncertainty. ECSQARU 2019. Lecture Notes in Computer Science(), vol 11726. Springer, Cham. https://doi.org/10.1007/978-3-030-29765-7_33
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